Differential operators on the base affine space of $S{L}_n$ and quantized Coulomb branches

Abstract

We show that the algebra $D_{\hslash }(S{L}_n/U)$ of differential operators on the base affine space of $S{L}_n$ is the quantized Coulomb branch of a certain 3d $\mathcal{N}=4$ quiver gauge theory. In the semiclassical limit this proves a conjecture of Dancer-Hanany-Kirwan about the universal hyperkähler implosion of $S{L}_n$. We also formulate and prove a generalization identifying the Hamiltonian reduction of $T^{\ast }S{L}_n$ with respect to an arbitrary unipotent character as a Coulomb branch. As an application of our results, we provide a new interpretation of the Gelfand-Graev symmetric group action on $D_{\hslash }(S{L}_n/U)$.

20 pages

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